Blood volume, V_B, in a human can be determined using the equation V_B = V_P/(1 - H), where V_P is...

1. TRANSLATE the problem requirements

• Given: \(\mathrm{VB} = \frac{\mathrm{VP}}{(1 - \mathrm{H})}\)
• Find: Express H in terms of VB and VP
• This means we need to solve for H and get it alone on one side

2. SIMPLIFY by eliminating the fraction

• Start with: \(\mathrm{VB} = \frac{\mathrm{VP}}{(1 - \mathrm{H})}\)
• Multiply both sides by (1 - H): \(\mathrm{VB}(1 - \mathrm{H}) = \mathrm{VP}\)
• This removes the fraction from the right side

3. SIMPLIFY to isolate terms with H

• From \(\mathrm{VB}(1 - \mathrm{H}) = \mathrm{VP}\), divide both sides by VB
• Result: \((1 - \mathrm{H}) = \frac{\mathrm{VP}}{\mathrm{VB}}\)
• Now H is only in one term on the left side

4. SIMPLIFY to solve for H

• From \((1 - \mathrm{H}) = \frac{\mathrm{VP}}{\mathrm{VB}}\), subtract 1 from both sides
• Result: \(-\mathrm{H} = \frac{\mathrm{VP}}{\mathrm{VB}} - 1\)
• Multiply both sides by -1: \(\mathrm{H} = 1 - \frac{\mathrm{VP}}{\mathrm{VB}}\)

Answer: A. \(\mathrm{H} = 1 - \frac{\mathrm{VP}}{\mathrm{VB}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors during the algebraic manipulation, particularly when dealing with the negative sign in front of H.

When subtracting 1 from both sides of \((1 - \mathrm{H}) = \frac{\mathrm{VP}}{\mathrm{VB}}\), students might write \(-\mathrm{H} = -1 + \frac{\mathrm{VP}}{\mathrm{VB}}\) instead of \(-\mathrm{H} = \frac{\mathrm{VP}}{\mathrm{VB}} - 1\), or make errors when multiplying by -1 at the end. These sign errors can lead to incorrect expressions.

This may lead them to select Choice C (\(\mathrm{H} = 1 + \frac{\mathrm{VB}}{\mathrm{VP}}\)) due to sign confusion.

Second Most Common Error:

Poor algebraic manipulation: Students might try to solve for H without properly clearing the fraction first, or make errors in the order of operations.

Some students might incorrectly think they can directly get \(\mathrm{H} = \frac{\mathrm{VB}}{\mathrm{VP}}\) from the original equation by misunderstanding the relationship between the variables.

This may lead them to select Choice B (\(\mathrm{H} = \frac{\mathrm{VB}}{\mathrm{VP}}\)).

The Bottom Line:

This problem requires careful, systematic algebraic manipulation with attention to signs and proper sequence of operations. Success depends on methodically working through each step rather than trying to take shortcuts.